FIRST-ORDER SYSTEM LEAST SQUARES FOR THE STOKES
               EQUATIONS, WITH APPLICATION TO LINEAR ELASTICITY

                                    Z. Cai
                               T. A. Manteuffel
                               S. F. McCormick

                                   Abstract

Following our earlier work on general second-order scalar equations, here we
develop a least-squares functional for the two- and three-dimensional Stokes
equations, generalized slightly by allowing a pressure term in the continuity
equation.  By introducing a velocity flux variable and associated curl and
trace equations, we are able to establish ellipticity in an H^1 product norm
appropriately weighted by the Reynolds number.  This immediately yields
optimal discretization error estimates for finite element spaces in this norm
and optimal algebraic convergence estimates for m ultiplicative and additive
multigrid methods applied to the resulting discrete systems.  Both estimates
are uniform in the Reynolds number.  Moreover, our pressure-perturbed form of
the generalized Stokes equations allows us to develop an analogous result for
the Dirichlet problem for linear elasticity, with estimates that are uniform
in the Lam{\'e} constants.